Lecture 5:
Anomalies, Risk, and Factor Models

Recap: Evidence of Anomalies

Behavioral finance excels at documenting many anomalies that challenge the traditional view of efficient markets.

• These anomalies suggest that investors may not always act rationally, leading to predictable patterns in stock returns.

We have seen evidence that certain strategies earn abnormal returns:

• Momentum: Buy past winners, sell past losers → positive returns
• Contrarian: Buy past losers, sell past winners → positive returns (longer horizons)

Harvey, Liu, and Zhu (2016) document that “…there are now more than 300 documented anomalies.”

These strategies use long-short portfolios to isolate the effect.

Key question: Are these returns compensation for risk, or evidence of market inefficiency caused by some behavioral bias?

The Zoo of Anomalies

More Anomalies and Their Behavioral Roots

Anomaly or Risk Factor?

When we observe that a strategy earns positive returns, two interpretations:

Risk-Based Explanation:

• The strategy is exposed to systematic risk
• Returns are compensation for bearing that risk
• Investors require higher returns for holding risky assets, i.e., strategy returns covary with bad economic states, generating losses when marginal utility is high
• I.e., You can’t diversify away this risk, so you get paid for bearing it.

Behavioral/Mispricing Explanation:

• Markets are inefficient
• Investors make systematic errors
• Returns are achievable until arbitraged away

How to Distinguish Risk from Mispricing?

Evidence for RISK:

• ✓ Returns covary with bad economic states (recessions, crises)
• ✓ Strategy is painful to hold when you need money most
• ✓ Persistent across time periods and markets
• ✓ Clear economic rationale for the risk

Evidence for MISPRICING:

• ✓ Linked to documented behavioral biases
• ✓ Limits to arbitrage prevent correction
• ✓ Varies with investor sentiment/attention
• ✓ Diminishes after publication (arbitraged away)

Testing Anomalies: The Methodology

Step 1: Form long-short portfolios based on the characteristic

• Sort stocks into deciles/quintiles
• Go long top decile, short bottom decile
• Track returns over time

Step 2: Regress the long-short portfolio returns on factor models

R_{LS,t} - R_f = \alpha + \beta^\prime Factors + \epsilon_t

Step 3: Examine the alpha (intercept)

• Significant \alpha > 0, with t-statistic significant → anomaly not explained by model
• \alpha \approx 0 with t-statistic insignificant → returns explained by factor exposures

Factor Models: From CAPM to Multi-Factor

CAPM (1 factor): E(R_i) - R_f = \beta_i (E(R_m) - R_f)

Fama-French 3-Factor (1993): E(R_i) - R_f = \beta_{MKT}(R_m - R_f) + \beta_{SMB} \cdot SMB + \beta_{HML} \cdot HML

Each additional factor potentially “explains away” anomalies. You saw in your investment management course that the 3-factor model captures size premium (small-minus-big, SMB, size factor) and value premium (high-minus-low, HML, book-to-market) effects.

What Are These Factors?

All factors are long-short portfolios themselves!

Interpreting Regression Output

Example regression output for an hypothetical momentum strategy:

Interpretation:

• Alpha = 0.50%: Monthly abnormal return not explained by factors
• MKT-RF ≈ 0: Little market exposure (market neutral)
• SMB < 0: Tilts toward large caps
• HML < 0: Tilts toward growth (momentum loads negatively on value stocks)

What Does Alpha Tell Us?

Significant positive alpha:

• The anomaly generates returns beyond factor exposures
• Either: (1) missing risk factor, or (2) true mispricing

Alpha ≈ 0:

• Factor model “explains” the anomaly
• Returns are compensation for factor exposures

But: Explaining ≠ Understanding

• Saying “momentum is explained by HML loading” doesn’t tell us WHY

Real Example: Media Coverage Anomaly

Fang and Peress (2009) find that stocks with low media coverage earn higher returns.

Strategy: Long low-coverage stocks, short high-coverage stocks

Interpretation: Alpha shrinks but remains marginally significant. Some of the effect is explained by size/value/momentum, but not fully. This could be a risk factor (e.g., illiquidity) or mispricing (neglected stocks). It’s up to the authors to make the case.

The Joint Hypothesis Problem

Fama (1970): “Tests of market efficiency are joint tests of:

1. The efficiency hypothesis, AND
2. A particular asset-pricing model”

Implication: When we find significant alpha, we cannot tell if:

• Markets are inefficient (mispricing), OR
• Our model is wrong (missing risk factor)

This is the fundamental identification problem in empirical asset pricing.

The Joint Hypothesis in Practice

Scenario: You find a strategy with significant CAPM alpha.

Interpretation A (Mispricing):

• “CAPM is correct, markets are inefficient”
• Behavioral biases create exploitable patterns

Interpretation B (Risk):

• “Markets are efficient, CAPM is incomplete”
• The strategy captures a missing risk factor
• Add the factor to the model (SMB, HML, etc.)

Reality: We can never fully resolve this. The debate continues.

Do Anomalies Survive Publication?

If anomalies reflect mispricing, arbitrageurs should trade them away once known.

Key finding: Many anomalies weaken significantly after publication.

Implication: Post-publication decay supports the mispricing view—informed traders arbitrage away the profits.

Why Do Some Anomalies Persist?

Not all anomalies disappear. Persistence may indicate:

1. Limits to Arbitrage (Shleifer & Vishny, 1997)

• Transaction costs, short-selling constraints
• Idiosyncratic risk deters arbitrageurs
• Noise trader risk

2. Continued Behavioral Biases

• New investors keep making the same mistakes
• Institutional constraints prevent exploitation

3. Compensation for Risk

• If it’s truly a risk factor, returns should persist
• No “free lunch” to arbitrage away

Key insight: Anomalies that persist despite publication might suggest that they are more likely risk-based.

Summary: How to Think About Anomalies

1. Document the anomaly using long-short portfolios

2. Test against factor models to see if alpha persists

3. Consider both explanations:

   • Risk: Does it hurt in bad times? Economic rationale?
   • Mispricing: Behavioral bias? Limits to arbitrage?

4. Remember the joint hypothesis:

   • Rejection of a model ≠ market inefficiency
   • Could be a missing factor, i.e., you don’t have the right asset pricing model!

5. Beware of data mining: 300+ anomalies suggests many are spurious